Diagram of Geodesic Intervals in $C'(1/8)$-Groups

Abstract

Let $Γ$ be a group characterized by a small cancellation $C'(1/8)$-presentation. We study the structure of its Cayley graph, specifically geodesic intervals and digons. We prove that all geodesic segments $[x,y]$ of length $n$ can be covered by finitely many simple geodesic digons with pairwise $C'(1/8)$-short lower intersections whose lower sides lie on a common geodesic segment $[x,y]$. This is a generalization of a proposition of Kharlampovich-Sklinos (arXiv:2106.05461) for random groups. We then bound this resulting diagram of digons, showing that the total perimeter of its cells and its edge count is bounded by $n$, and the cycle rank is equal to the number of distinct cell contours. Thus, although there may be many different geodesics, the size of their union is bounded linearly in the distance between their endpoints. The constant is independent of the number and lengths of the defining relators. In particular, the estimate applies to infinite presentations with unbounded relator lengths. From this we get a consequence for growth, allowing us to compare the number of geodesic words with the number of group elements of the same length.